I'm a masters student currently deciding in which area should I focus on. So far my primary interest has been C* algebras and operator algebras (already have some knowledge on K-theory for C* algebras and Hilbert modules), but I always had some interest in geometry. When I learnt about the Gelfand representation theorem I thought there was a chance to mix both interests. I tried to read Alain Connes book I realized I was missing lots of prerequisites. I'd like to have at least a picture broad enough of this area to determine if I like it or not, what is the mininum I should know before I can do that?

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    $\begingroup$ Try reading this arxiv.org/abs/math/0408416 There is also a book by the same author which is much friendlier than Connes' book. I'd say given that you already know some Operator algebras and K-theory, probably some algebraic topology and differential geometry is useful. Try understanding the Atiyah Singer index theorem statement. $\endgroup$ – vap Nov 9 '17 at 19:23
  • $\begingroup$ I'm currently taking a course in diff geometry and took an introductory course for alg topology. Any book to study the Atiyah Singer index theorem? $\endgroup$ – Julio Cáceres Nov 10 '17 at 13:22
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    $\begingroup$ Heat Kernels and Dirac Operators Book by Ezra Getzler, Michèle Vergne, and Nicole Berline. The approach in this book is analytic. If you want a more K-theoretic approach there are two papers by Nigel Higson (One has K-theory in its name, the other deals with the tangemt groupoid) that you can google easily. $\endgroup$ – vap Nov 13 '17 at 2:41
  • $\begingroup$ Elliot Natsume Nest offer a different approach via the "classical limit". Trying to understand all of these proofs will keep you busy for a while and it'll get you to learn a lot around noncommutative geometry. $\endgroup$ – vap Nov 13 '17 at 2:50

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